Three real-space discretization techniques in electronic structure calculations

A characteristic feature of the state-of-the-art of real-space methods in electronic structure calculations is the diversity of the techniques used in the discretization of the relevant partial differential equations. In this context, the main approaches include finite-difference methods, various types of finite-elements and wavelets. This paper reports on the results of several code development projects that approach problems related to the electronic structure using these three different discretization methods. We review the ideas behind these methods, give examples of their applications, and discuss their similarities and differences.Comment: 39 pages, 10 figures, accepted to a special issue of "physica status solidi (b) - basic solid state physics" devoted to the CECAM workshop "State of the art developments and perspectives of real-space electronic structure techniques in condensed matter and molecular physics". v2: Minor stylistic and typographical changes, partly inspired by referee comment

A New Family of High Order Unstructured MOOD and ADER Finite Volume Schemes for Multidimensional Systems of Hyperbolic Conservation Laws

International audienceIn this paper, we investigate the coupling of the Multi-dimensional Optimal Order De- tection (MOOD) method and the Arbitrary high order DERivatives (ADER) approach in order to design a new high order accurate, robust and computationally efficient Finite Volume (FV) scheme dedicated to solve nonlinear systems of hyperbolic conservation laws on unstructured triangular and tetrahedral meshes in two and three space dimensions, respectively. The Multi-dimensional Optimal Order Detection (MOOD) method for 2D and 3D geometries has been introduced in a recent series of papers for mixed unstructured meshes. It is an arbitrary high-order accurate Finite Volume scheme in space, using polynomial reconstructions with a posteriori detection and polynomial degree decre- menting processes to deal with shock waves and other discontinuities. In the following work, the time discretization is performed with an elegant and efficient one-step ADER procedure. Doing so, we retain the good properties of the MOOD scheme, that is to say the optimal high-order of accuracy is reached on smooth solutions, while spurious oscillations near singularities are prevented. The ADER technique permits not only to reduce the cost of the overall scheme as shown on a set of numerical tests in 2D and 3D, but it also increases the stability of the overall scheme. A systematic comparison between classical unstructured ADER-WENO schemes and the new ADER-MOOD approach has been carried out for high-order schemes in space and time in terms of cost, robustness, accuracy and efficiency. The main finding of this paper is that the combination of ADER with MOOD generally outperforms the one of ADER and WENO either because at given accuracy MOOD is less expensive (memory and/or CPU time), or because it is more accurate for a given grid resolution. A large suite of classical numerical test problems has been solved on unstructured meshes for three challenging multi-dimensional systems of conservation laws: the Euler equations of compressible gas dynamics, the classical equations of ideal magneto-Hydrodynamics (MHD) and finally the relativistic MHD equations (RMHD), which constitutes a particularly challenging nonlinear system of hyperbolic par- tial differential equation. All tests are run on genuinely unstructured grids composed of simplex elements

Computational Electromagnetism and Acoustics

It is a moot point to stress the significance of accurate and fast numerical methods for the simulation of electromagnetic fields and sound propagation for modern technology. This has triggered a surge of research in mathematical modeling and numerical analysis aimed to devise and improve methods for computational electromagnetism and acoustics. Numerical techniques for solving the initial boundary value problems underlying both computational electromagnetics and acoustics comprise a wide array of different approaches ranging from integral equation methods to finite differences. Their development faces a few typical challenges: highly oscillatory solutions, control of numerical dispersion, infinite computational domains, ill-conditioned discrete operators, lack of strong ellipticity, hysteresis phenomena, to name only a few. Profound mathematical analysis is indispensable for tackling these issues. Many outstanding contributions at this Oberwolfach conference on Computational Electromagnetism and Acoustics strikingly confirmed the immense recent progress made in the field. To name only a few highlights: there have been breakthroughs in the application and understanding of phase modulation and extraction approaches for the discretization of boundary integral equations at high frequencies. Much has been achieved in the development and analysis of discontinuous Galerkin methods. New insight have been gained into the construction and relationships of absorbing boundary conditions also for periodic media. Considerable progress has been made in the design of stable and space-time adaptive discretization techniques for wave propagation. New ideas have emerged for the fast and robust iterative solution for discrete quasi-static electromagnetic boundary value problems

Proceedings of the FEniCS Conference 2017

Proceedings of the FEniCS Conference 2017 that took place 12-14 June 2017 at the University of Luxembourg, Luxembourg

Aeronautical engineering: A continuing bibliography with indexes (supplement 309)

This bibliography lists 212 reports, articles, and other documents introduced into the NASA scientific and technical information system in Oct. 1994. Subject coverage includes: design, construction and testing of aircraft and aircraft engines; aircraft components, equipment, and systems; ground support systems; and theoretical and applied aspects of aerodynamics and general fluid dynamics

Design, Analysis and Fabrication of Silicon-Based Optical Materials and Photonic Crystal Devices

As the integration of electronic components grow so does the need for low power, low cost, and high-speed devices. These have resulted in an increased need for complementary metal-oxide semiconductor (CMOS) compatible materials and fabrication technique for novel structures as well as accurate models of the electromagnetic eld behavior in them. Recent advances in materials technology and fabrication techniques have made it feasible to consider silicon (Si)-based optical materials and photonic crystal (PhC) de- vices having physical dimensions of the order of the optical wavelength as the possible means to achieve these needs. Research has shown that light emission from Si is possible in low-dimensional state, i.e., Si-nanocrystals (Si-ncs). Furthermore, three-dimensional (3-D) control of light compatible with CMOS fabrication technology is required in order to fully integrate optical functionalities into the existing Si-technology. However, the di - culties in the fabrication of 3-D PhC waveguides have resulted in using two-dimensional (2-D) PhC structures. Finally, numerical simulations pro- vide a framework for quick low-cost feasibility studies and allow for design optimization before devices are fabricated. In this dissertation, we present our e orts along these directions. This dissertation addressed the method of obtaining high quantum e ciency from Si-ncs compatible with CMOS processing. Si ions were implanted into a fused-silica substrate (10 mm 10 mm 1 mmt) at room temperature in the Takasaki ion accelerators for advanced radiation application (TIARA) of the Japan Atomic Energy Agency. The implantation energy was 80 keV, and the implantation amount was 2 1017 ions/cm2. The Si-implanted sub- strate was cut into four pieces (5 mm 5 mm 1 mmt) using a diamond-wire saw, and the four pieces were annealed in ambient air at 1100, 1150, 1200, and 1250 oC for 25 min in a siliconit furnace. PL spectra were measured at room temperature with excitation using a He-Cd laser ( =325 nm). Ultra- violet (UV)-PL spectra having peaks around a wavelength of 370 nm were observed from all the samples. In our experiments, the UV-PL peak had a maximum intensity after annealing at 1250 oC, and the longer wavelength PL peak around 800 nm observed from the samples annealed at 1100 and 1150 oC disappeared by annealing above 1200 oC. The two PL peaks of the Si-ion-implanted samples may have originated from interface layers be- tween Si-ncs and SiO2 media. However, we successfully obtained only the UV-light emission peaks by selecting the proper annealing temperatures. UV-light-emitting materials are expected to be useful as light sources for next-generation optical-disk systems whose data densities are higher than Blu-ray Disk systems. Additionally, this dissertation addressed the numerical modeling of PhC de- vices. Accurate computations can provide a detailed understanding of the complex physical phenomena inherent in PhC devices. The nite-di erence time-domain (FDTD) method, which is widely used by many researchers around the Globe, is a powerful tool for modeling PhC devices. We devel- oped a modi ed and easy FDTD method based on a regular Cartesian Yee's lattice for calculating the dispersion diagram of triangular lattice PhCs. Our method uses the standard central-di erence equation, which is very easy to implement in any computing environment. The Bloch periodic boundary conditions are applied on the sides of the unit cell by translating the periodic boundary conditions to match with the directions of periodicity in the tri- angular lattice. Complete and accurate bandgap information is obtained by using this FDTD approach. Convergence, accuracy, and stability analysis were carried out, which ensures the reliability of this method. Numeri- cal results for 2-D transverse electric (TE) and transverse magnetic (TM) modes in triangular lattice PhCs are in good agreement with results from 2-D plane wave expansion method. The obtained results are in consistence with the reported ones. To ease the practical application of this method, clear explanations on the computer implementation are also provided. Finally, this dissertation addressed the use of CMOS-compatible fabrication method and 2-D periodic structures to realize the control of light in 3-D. In particular, we designed, analyzed and fabricated novel PhC waveguides utilizing Si-ion implantation and 2-D periodic structures. The transport of ions in matter (TRIM) prediction of implantation depth distribution pro le (1 1017 ions/cm2, 80 keV) shows the range of about 150 nm. Assuming the e ective refractive index of the Si-rich region to be 1.89 and by using FDTD method, the PhC design parameters based on the telecommunication wave- length ( =1.55 m) were obtained by varying the radius to lattice constant ratio (r=a) from 0.2 to 0.45. We analyzed both TE and TM mode prop- agation in triangular-lattice PhCs. The designed parameters were found to be a=664 nm and r=a=0.35. The PBG spanned from normalized fre- quency of 0.39 to 0.46 [2 c/a] in the TE-mode triangular lattice and the gap to midgap ratio was 0.16. The designed pattern was fabricated and the diameter, the period and the depth of air holes of the waveguide were estimated by atomic force microscopy (AFM) to be 464, 666 and 175 nm, respectively. Numerical results using FDTD characterization show that, straight line PhC waveguides can achieve 100% transmission, while the 60o bend showed 80% transmission owing to the dispersion mismatch at the two 60o bends. These results may serve as useful guides and components in future high- density photonic integrated circuits associated with optical communications, computing, and signal processing.学位記番号：工博甲40